## Gauss Elimination Method

DEFINITION 2.2.10 (Forward/Gauss Elimination Method)   Gaussian elimination is a method of solving a linear system (consisting of equations in unknowns) by bringing the augmented matrix

to an upper triangular form

This elimination process is also called the forward elimination method.

The following examples illustrate the Gauss elimination procedure.

EXAMPLE 2.2.11   Solve the linear system by Gauss elimination method.

Solution: In this case, the augmented matrix is The method proceeds along the following steps.
1. Interchange and equation (or ).

2. Divide the equation by (or ).

3. Add times the equation to the equation (or ).

4. Add times the equation to the equation (or ).

5. Multiply the equation by (or ).

The last equation gives the second equation now gives Finally the first equation gives Hence the set of solutions is A UNIQUE SOLUTION.

EXAMPLE 2.2.12   Solve the linear system by Gauss elimination method.

Solution: In this case, the augmented matrix is and the method proceeds as follows:
1. Add times the first equation to the second equation.

2. Add times the first equation to the third equation.

3. Add times the second equation to the third equation

Thus, the set of solutions is with arbitrary. In other words, the system has INFINITE NUMBER OF SOLUTIONS.

EXAMPLE 2.2.13   Solve the linear system by Gauss elimination method.

Solution: In this case, the augmented matrix is and the method proceeds as follows:
1. Add times the first equation to the second equation.

2. Add times the first equation to the third equation.

3. Add times the second equation to the third equation

The third equation in the last step is

This can never hold for any value of Hence, the system has NO SOLUTION.

Remark 2.2.14   Note that to solve a linear system, one needs to apply only the elementary row operations to the augmented matrix

A K Lal 2007-09-12